# Fun Ellipse Problem

1. Feb 10, 2010

### Seebs

This is a small part of a larger program I'm working on. This actually looked like a fun problem -- but I'm hitting a wall now.

Imagine a person riding a bicycle. You know their starting position (X1, Y1), and their initial heading.
Their destination is elsewhere at point (X2, Y2). They MUST ride a total distance of S along an elliptical path to reach their destination.
Assuming S is a sufficiently large enough value to actually travel to the destination, find the parametric representation of the path traveled.

What I've got so far:
The parametric representation of a general ellipse is as follows:
X(t) = Xc + a cos(t) cos($$\phi$$) - b sin(t) sin($$\phi$$)
Y(t) = Yc + a cos(t) sin($$\phi$$) + b sin(t) cos($$\phi$$)

where Xc, and Yc are the center of the ellipse, a and b are the major and minor semi-axes respectively, and $$\phi$$ is the angle between the X-axis and the major axis ( http://en.wikipedia.org/wiki/Ellipse#General_parametric_form" )

I would need to solve for Xc, Yc, a, b, $$\phi$$, t1, and t2.
7 unknowns, so I'd need 7 equations.

I get 4 equations using the start and end positions.
Since I know the heading at the start point, if I take the derivative of the general parametric equations I can use the slope of the starting point to get 2 more equations.
And finally, I can use my travel distance, S, with the general parametric arc length equation. http://tutorial.math.lamar.edu/Classes/CalcII/ParaArcLength.aspx" [Broken]

If I clean it up, I get this equation for the arc length between t1 and t2 on an ellipse (excuse the formatting)
S = $$\int$$$$\stackrel{t2}{t1}$$( a^2 sin(t)^2 + b^2 cos(t)^2 ) dt

So here's my wall.
I am implementing this into a computer program, and would need to find "a good solution" quickly. That elliptical arc length equation is throwing me a monkey wrench. Without it, I believe I could find the Jacobian, use Newton's method, and get my 7 unknowns in a handful of iterations.

I guess my question is, am I on the right track here? What is the best way to proceed? Is there a "why didn't you just" solution I'm not seeing?

(In my gut, something doesn't feel right -- like my two "heading" equations are not really independent of each other. I also think I can visualize at least two solutions to this problem given the set-up. If this is the case, that there isn't ONE solution, I'd simply need A solution. Any thoughts?)

Last edited by a moderator: May 4, 2017